Ratio of geometirc means

The lognormal t test compares the geometric means of two groups. If the geometric means of both groups are the same, then this ratio will equal 1. If one group is larger than the other, then this ratio will be either greater or less than 1, depending on the direction that the ratio is defined.

The most useful result is the confidence interval for the ratio of the geometric means. The point of the experiment was to determine if the geometric means of the groups were the same. The confidence interval tells you how precisely you know that the ratio of the geometric means. If the assumptions of the analysis are true, you can be 95% confident that the 95% confidence interval contains the true ratio between the geometric means.

For many purposes, this confidence interval is all you need. Note that you can change the "direction" of the ratio in the Options tab of the analysis dialog, where you can tell Prism to calculate the ratio as B/A or A/B. Note that this will cause the ratio and the 95% confidence interval to be different, but the P value will remain unchanged.

P value

The P value is used to ask whether the ratio of the geometric means for the two groups is likely to be due to chance. It answers this question:

If the two populations really had the same geometric mean, what is the chance that random sampling would result in a ratio of geometric means as extreme (or more so) than observed in this experiment?

"Extreme" in this case refers to how far from the null value of 1.0 the ratio of geometric means is. Small P values could be obtained with ratios that are either smaller or larger than 1.0. This is why changing the direction of the ratio (B/A vs A/B) can change the 95% confidence interval, but not the P value.

It is traditional, but not necessary and often not useful, to use the P value to make a simple statement about whether or not the difference is “statistically significant”.

You will interpret the results differently depending on whether the P value is small or large.

F test compare variances

The lognormal t test relies on the assumption that the variance of the log-transformed populations from which the data were sampled are equal. This is equivalent to saying that the geometric standard deviations of the two populations are the same, but NOT the same as saying that the variances of the two populations are the same (two lognormally distributed populations can have the same geometric standard deviation and different variances). Prism tests this assumption using an F test.

First compute the geometric standard deviations of both groups, take the logarithm of each, then square them both to obtain variances on the log scale. The F ratio equals the larger variance divided by the smaller variance. So F is always greater than (or possibly equal to) 1.0.

The P value then asks:

If the two populations really had identical variances, what is the chance of obtaining an F ratio this big or bigger?

Don't mix up the P value testing for equality of the variances of the groups with the P value testing for equality of the geometric means. That latter P value is the one that answers the question you most likely were thinking about when you chose the t test.

What to do when the groups have different variances?

R squared from unpaired t test

Prism, unlike most statistics programs, reports a R2 value as part of the unpaired t test results. It quantifies the fraction of all the variation in the samples that is accounted for by a difference between the group means. If R2=0.36, that means that 36% of all the variation among values is attributed to differences between the two group means, leaving 64% of the variation that comes from scatter among values within the groups.

If the two groups have the same mean, then none of the variation between values would be due to differences in group means so R2 would equal zero. If the difference between group means is huge compared to the scatter within the group, then almost all the variation among values would be due to group differences, and the R2 would be close to 1.0.

Additional resources

This (somewhat long) article explains how to think about and analyze lognormal data. It includes specific examples for performing and analyzing lognormal t tests:

HJ Motulsky, T Head, PBS Clarke, 2025, Analyzing Lognormal Data: A Nonmathematical Practical Guide. Pharmacological Reviews, Volume 77, Issue 3, 100049